Educational model. Seeds along the strip grow from small (left) to large (right); each finch species eats a bell-shaped range of seed sizes set by its beak. Curtains show each species' niche; the grey curtain at the back is seed supply. Drag to orbit, wheel or pinch to zoom.

Finch Beaks, Niche Overlap and Competitive Exclusion

Move the species along the seed-size gradient and watch Lotka-Volterra competition decide who coexists, who is excluded, and how beaks can drift apart.

Time

Species (preferred seed size)

Niche geometry and growth

Live readouts

Population over time

Niches along the seed gradient

Preferred seed size over time

Experiment: how wide may niches be before one species is excluded?

Species 1 sits at the resource peak (seed size 0), species 2 at the separation set by your slider. Sweeps the niche width from narrow to wide and finds where species 2 disappears.

Independent numerical checks

Controls

Drag or one finger: orbit; wheel or pinch: zoom; Space: pause.
Sliders edit the live model; the third species is optional. Evolution speed above zero lets beaks shift toward higher growth (character displacement).

The science

Competition model. For species i with density N_i: dN_i/dt = r N_i (1 - sum_j a_ij N_j / K_i). The competition coefficient comes from niche overlap: if species use seed size x with Gaussian utilisation curves of width s, a_ij = exp(-(x_i - x_j)^2 / (4 s^2)) (the overlap integral of two Gaussians divided by the self-overlap). Seed supply is Gaussian with width sK, so a species centred at x_i has carrying capacity K_i = K0 exp(-x_i^2 / (2 sK^2)).

Coexistence rule. For two species, both persist if and only if a_12 < K_2/K_1 and a_21 < K_1/K_2. If species 1 sits at the resource peak and species 2 at distance d, this reduces to exp(-d^2/4s^2) < exp(-d^2/2sK^2), i.e. s < sK / sqrt(2) regardless of d: narrow niches coexist, wide ones lose the weaker species (competitive exclusion, Gause's principle). The page tests this threshold numerically.

Equilibrium. When all species persist, sum_j a_ij N_j* = K_i, a linear system solved exactly by Gaussian elimination and compared with the long-time Runge-Kutta result. A lone species obeys the logistic solution N(t) = K / (1 + (K/N0 - 1) e^(-r t)), also checked.

Character displacement. With evolution speed mu above zero, each preferred seed size moves uphill on its own per-capita growth rate, dx_i/dt = mu df_i/dx_i with f_i = r(1 - sum_j a_ij N_j / K_i). Competitors then push each other apart while the resource peak pulls them inward, the textbook pattern seen in Darwin's finches. This is an adaptive-dynamics caricature, not a genetic model: no mutation, no sexual reproduction, no drift.

Honest simplifications. Gaussian niches and resource supply; densities are continuous; one resource axis (seed size) and no seasons, predators, or seed regrowth dynamics; all species share one growth rate r; time is in units of 1/r, not days. Beak sizes in the 3D models are stylised and drawn larger than life; the drawn bird counts are proportional to density (about 1 bird per 4 units) and positions are quantiles of each utilisation curve, not individual tracking. Real finch ecology (for example the Galapagos record) is more complicated.

Model and assets: Blender (island, finches, seeds, acacia) + Three.js. Independent educational model.